The sum of two angles is $84^\circ$. Angle 2 is $42^\circ$ smaller than $2$ times angle 1. What are the measures of the two angles in degrees?
Answer: Let $x$ equal the measure of angle 1 and $y$ equal the measure of angle 2. The system of equations is then: ${x+y = 84}$ ${y = 2x-42}$ Since we already have solved for $y$ in terms of $x$ , we can use substitution to solve for $x$ and $y$ Substitute ${2x-42}$ for $y$ in the first equation. ${x + }{(2x-42)}{= 84}$ Simplify and solve for $x$ $ x+2x - 42 = 84 $ $ 3x-42 = 84 $ $ 3x = 126 $ $ x = \dfrac{126}{3} $ ${x = 42}$ Now that you know ${x = 42}$ , plug it back into $ {y = 2x-42}$ to find $y$ ${y = 2}{(42)}{ - 42}$ $y = 84 - 42$ ${y = 42}$ You can also plug ${x = 42}$ into $ {x+y = 84}$ and get the same answer for $y$ ${(42)}{ + y = 84}$ ${y = 42}$ The measure of angle 1 is $42^\circ$ and the measure of angle 2 is $42^\circ$.